This week the question is: Can You Survive March Madness?

March Madness—the NCAA’s men’s and women’s basketball tournaments—is here!

The single-elimination tournament brackets consists of 64 teams spread across four regions, each with teams seeded 1 through 16. (In recent years, additional teams beyond the 64 have been added, but you needn’t worry about these teams for this week’s puzzle.)

Among the 16 teams in a region, you might wonder which team has the toughest schedule. One way to evaluate a team’s strength of schedule within the region is to compute the geometric mean of strongest opponents a team can face in various rounds.

For example, the 1-seed faces the 16-seed in the first round, then (potentially) the 8-seed in the second round, then (potentially) the 4-seed in the third round, and finally (potentially) the 2-seed in the fourth round. The geometric mean of these opponents is the fourth root (since there are four opponents) of 16 · 8 · 4 · 2, or approximately 5.66. In this computation, we used the 8-seed rather than the 9-seed because 8 is less than 9, we used the 4 seed rather than the 5-seed, 12-seed, or 13-seed because 4 is less than 5, 12, and 13, and so on. The tougher a team’s strength of schedule, the lower this geometric mean.

Of the 16 teams in the region, which two seeds have the toughest strength of schedule?

And for extra credit:

Instead of 16 seeded teams in a region, suppose there are 2^N seeded teams in the region, where N is a very, very large number.

The two seeds with the toughest strengths of schedule have seeds that approach fractional values of 2^N. What are these two fractions?

And here’s my suggested solution. Including my reasoning.